科目：进阶数学 Further Mathematics (9231)
年份：2009 冬季
试卷：Paper 2
总页数：7
章节：9231 Further Mathematics
1. A particle of mass $m$ is attached to one end $A$ of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$ and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of $120^\circ$. Show that the speed of the particle is then $\sqrt{\left(\frac{1}{2}ga\right)}$ and find the initial speed of projection.
2. A circular wheel is modelled as a uniform disc of mass $6\text{ kg}$ and radius $0.25\text{ m}$. It is rotating with angular speed $2\text{ rad/s}^{-1}$ about a fixed smooth axis perpendicular to its plane and passing through its centre. A braking force of constant magnitude is applied tangentially to thTwo uniform rods, $AB$ and $BC$, each have length $2a$ and weight $W$. They are smoothly jointed at $B$, and $A$ is attached to a smooth fixed pivot. A light inextensible string of length $(2\sqrt{2})a$ joins $A$ to $C$ so that angle $ABC = 90^\circ$. The system hangs in equilibrium, with $AB$ making an angle $\alpha$ with the vertical (see diagram). By taking moments about $A$ for the system, or otherwise, show that $\alpha = 18.4^\circ$, correct to the nearest 0.1°.
Find the tension in the string in the form $kW$, giving the value of $k$ correct to 3 significant figures.
Find, in terms of $W$, the magnitude of the force acting on the rod $BC$ at $B$.
6 A machine produces metal discs whose diameters have a normal distribution. The mean of this distribution is intended to be 10 cm8 150 sheep, chosen from a large flock of sheep, were divided into two groups of 75. Over a fixed period, one group had their grazing controlled and the other group grazed freely. The gains in weight, in kg, were recorded for each animal and the table below shows the sample means and the unbiased estimates of the population variances for the two samples. | | Sample mean | Unbiased estimate of population variance | | :--- | :---: | :---: | | Controlled grazing | 19.14 | 20.54 | | Free grazing | 15.36 | 9.84 | It is required to test whether the population mean for sheep having their grazing controlled exceeds the population mean for sheep grazing freely by less than 5 kg. State, giving a reason, if it is necessary for the validity of the test to assume that the two population varian
11 Answer only one of the following two alternatives.
EITHER
A light elastic string, of natural length $l$ and modulus of elasticity $4mg$, is attached at one end to a fixed point and has a particle $P$ of mass $m$ attached to the other end. When $P$ is hanging in equilibrium under gravity it is given a velocity $\sqrt{gl}$ vertically downwards. At time $t$ the downward displacement of $P$ from its equilibrium position is $x$. Show that, while the string is taut,
$$\ddot{x} = -\frac{4g}{l}x.$$
[4]
Find the speed of $P$ when the length of the string is $l$. [4]
Show that the time taken for $P$ to move from the lowest point to the highest point of its motion is
$$\left(\frac{\pi}{3} + \frac{\sqrt{3}}{2}\right)\sqrt{\frac{l}{g}}.$$
[6]
OR
The scatter diagram BLANK PAGE
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